Density of pythagorean triplets – Part 2

Several people really liked my previous puzzle, including some high-school students 🙂 Some of you have sent me photos of your hand-written proofs. This level of enthusiasm for mathematics is totally awesome. Here is a detailed proof with some beautiful diagrams by Chris Grossack.

Here is my next puzzle:

Let x_1 < x_2 and a^2 + {x_1}^2 = {y_1}^2 and a^2 + {x_2}^2 = {y_2}^2.

Let x_3 < x_4 and b^2 + {x_3}^2 = {y_3}^2 and b^2 + {x_4}^2 = {y_4}^2.

Also, x_2 - x_1 = x_4 - x_3 = \delta

All the above variables are positive integers and a \neq b and x_1 \neq x_3 and x_2 \neq x_4

Prove or disprove the following claim:

Claim: There exists a rational number 0 < q < \delta such that \sqrt{a^2 + {(x_1 + q)}^2} and \sqrt{b^2 + {(x_3 + q)}^2} are rational numbers.

If you want to take small baby steps towards a proof, start with the following special cases:

Special case 1: x_1 = x_3 and x_2 = x_4

Special case 2: x_1 = x_3 and x_2 = x_4 and a = b

Quick homework problem: Prove the Special case 2 using a proof of my previous puzzle.

Have fun solving.

Some mathematician has said pleasure lies not in discovering truth, but in seeking it 🙂

Density of pythagorean triplets

I designed this simple math puzzle last week:

Prove or disprove the following statement:

  • Claim: Let p, q, r, s be positive integers such that p < q and r < s. Given two rational numbers \frac{p}{q} < \frac{r}{s}, there exist positive integers a, b such that a < b and \frac{p}{q} < \frac{a}{b} < \frac{r}{s} and a^2 + b^2 is a perfect square.
  • If the above statement is true, show an example of a, b, expressed in terms of p, q, r, s.
  • If the above statement is false, construct a counterexample.

Leave your solutions in the comments. Have fun solving it.